They both aim to be JavaScript supersets, so in theory all your code should already work. In practice some of the things you're currently doing will probably upset the type checks. Nothing particularly hard to fix, and you could only use them on new files, or move files over gradually.
If you're targetting ES5 with Babel, you can have TypeScript output ES6, and then pass that through babel. All libs should work fine, but you will not get all the benefits of the type checking if you don't install type definitions for them (some libs already come with those, though).
There really isn't a lot to lose. If you ever feel like it was not a good fit for your project, you can run your code though TypeScript, targetting ES6, and it will produce pretty much the same code without the type annotations.
If you're targetting ES5 with Babel, you can have TypeScript output ES6, and then pass that through babel. All libs should work fine, but you will not get all the benefits of the type checking if you don't install type definitions for them (some libs already come with those, though).
There really isn't a lot to lose. If you ever feel like it was not a good fit for your project, you can run your code though TypeScript, targetting ES6, and it will produce pretty much the same code without the type annotations.